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I am talking about a generalization of this inequality posed in this question.

let $x,y,z>0$ and such $xyz=1$, show that $$x^3+y^3+z^3+3\ge 2(x^2+y^2+z^2)$$

I am trying to prove the following inequality,

$$\displaystyle\sum_{i=1}^{n(n-2)} x_i^{n}-(n-1)\displaystyle\sum_{i=1}^n x_i^{n-1} \geq-n$$

where $x_i\in \mathbb{R^+}$ and $\displaystyle\prod_{i=1}^n x_i=1$.

The methods used to solve the linked inequality doesn't seem to be applied in this general case. Can anyone help me in proving it?

Community
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2 Answers2

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Maybe you mean the following inequality?

Let $x_1$, $x_2$, ..., $x_n$ be positive numbers such that $x_1x_2...x_n=1$. Prove that: $$(n-2)(x_1^n+x_2^n+...+x_n^n)-(n-1)\left(x_1^{n-1}+x_2^{n-1}+...+x_n^{n-1}\right)+n\geq0$$

Michael Rozenberg
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I wish to use lagrange multipliers: $$f(x,y,z)=\sum x^n-(n-1)\sum x^{n-1}+n\\ g(x,y,z)=\prod x=1$$ Now: $$nx^{n-1}-(n-1)^2x^{n-2}=\lambda(1/x)\implies \lambda=nx^n-(n-1)^2x^{n-1}$$ One such acceptable solution is $x=y=z=1$ where $f(x)=0$

RE60K
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